author | wenzelm |
Sun, 09 Apr 2006 18:51:13 +0200 | |
changeset 19380 | b808efaa5828 |
parent 18648 | 22f96cd085d5 |
permissions | -rw-r--r-- |
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(* Title: Parity.thy |
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ID: $Id$ |
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Author: Jeremy Avigad |
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*) |
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header {* Even and Odd for ints and nats*} |
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|
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theory Parity |
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imports Divides IntDiv NatSimprocs |
15131 | 10 |
begin |
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|
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axclass even_odd < type |
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|
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consts |
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even :: "'a::even_odd => bool" |
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instance int :: even_odd .. |
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instance nat :: even_odd .. |
|
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|
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defs (overloaded) |
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even_def: "even (x::int) == x mod 2 = 0" |
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even_nat_def: "even (x::nat) == even (int x)" |
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abbreviation |
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odd :: "'a::even_odd => bool" |
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"odd x == \<not> even x" |
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||
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|
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subsection {* Even and odd are mutually exclusive *} |
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|
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lemma int_pos_lt_two_imp_zero_or_one: |
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"0 <= x ==> (x::int) < 2 ==> x = 0 | x = 1" |
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by auto |
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|
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lemma neq_one_mod_two [simp]: "((x::int) mod 2 ~= 0) = (x mod 2 = 1)" |
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apply (subgoal_tac "x mod 2 = 0 | x mod 2 = 1", force) |
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apply (rule int_pos_lt_two_imp_zero_or_one, auto) |
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done |
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|
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subsection {* Behavior under integer arithmetic operations *} |
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|
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lemma even_times_anything: "even (x::int) ==> even (x * y)" |
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by (simp add: even_def zmod_zmult1_eq') |
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|
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lemma anything_times_even: "even (y::int) ==> even (x * y)" |
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by (simp add: even_def zmod_zmult1_eq) |
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|
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lemma odd_times_odd: "odd (x::int) ==> odd y ==> odd (x * y)" |
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by (simp add: even_def zmod_zmult1_eq) |
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|
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lemma even_product: "even((x::int) * y) = (even x | even y)" |
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apply (auto simp add: even_times_anything anything_times_even) |
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apply (rule ccontr) |
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apply (auto simp add: odd_times_odd) |
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done |
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|
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lemma even_plus_even: "even (x::int) ==> even y ==> even (x + y)" |
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by (simp add: even_def zmod_zadd1_eq) |
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|
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lemma even_plus_odd: "even (x::int) ==> odd y ==> odd (x + y)" |
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by (simp add: even_def zmod_zadd1_eq) |
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|
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lemma odd_plus_even: "odd (x::int) ==> even y ==> odd (x + y)" |
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by (simp add: even_def zmod_zadd1_eq) |
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|
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lemma odd_plus_odd: "odd (x::int) ==> odd y ==> even (x + y)" |
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by (simp add: even_def zmod_zadd1_eq) |
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|
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lemma even_sum: "even ((x::int) + y) = ((even x & even y) | (odd x & odd y))" |
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apply (auto intro: even_plus_even odd_plus_odd) |
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apply (rule ccontr, simp add: even_plus_odd) |
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apply (rule ccontr, simp add: odd_plus_even) |
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done |
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|
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lemma even_neg: "even (-(x::int)) = even x" |
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by (auto simp add: even_def zmod_zminus1_eq_if) |
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|
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lemma even_difference: |
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"even ((x::int) - y) = ((even x & even y) | (odd x & odd y))" |
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by (simp only: diff_minus even_sum even_neg) |
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|
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lemma even_pow_gt_zero [rule_format]: |
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"even (x::int) ==> 0 < n --> even (x^n)" |
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apply (induct n) |
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apply (auto simp add: even_product) |
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done |
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|
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lemma odd_pow: "odd x ==> odd((x::int)^n)" |
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apply (induct n) |
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apply (simp add: even_def) |
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apply (simp add: even_product) |
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done |
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|
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lemma even_power: "even ((x::int)^n) = (even x & 0 < n)" |
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apply (auto simp add: even_pow_gt_zero) |
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apply (erule contrapos_pp, erule odd_pow) |
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apply (erule contrapos_pp, simp add: even_def) |
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done |
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|
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lemma even_zero: "even (0::int)" |
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by (simp add: even_def) |
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|
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lemma odd_one: "odd (1::int)" |
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by (simp add: even_def) |
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|
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lemmas even_odd_simps [simp] = even_def[of "number_of v",standard] even_zero |
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odd_one even_product even_sum even_neg even_difference even_power |
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108 |
|
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109 |
|
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subsection {* Equivalent definitions *} |
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|
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lemma two_times_even_div_two: "even (x::int) ==> 2 * (x div 2) = x" |
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by (auto simp add: even_def) |
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|
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lemma two_times_odd_div_two_plus_one: "odd (x::int) ==> |
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2 * (x div 2) + 1 = x" |
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apply (insert zmod_zdiv_equality [of x 2, THEN sym]) |
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by (simp add: even_def) |
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|
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lemma even_equiv_def: "even (x::int) = (EX y. x = 2 * y)" |
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apply auto |
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apply (rule exI) |
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by (erule two_times_even_div_two [THEN sym]) |
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|
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lemma odd_equiv_def: "odd (x::int) = (EX y. x = 2 * y + 1)" |
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apply auto |
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apply (rule exI) |
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by (erule two_times_odd_div_two_plus_one [THEN sym]) |
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|
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|
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subsection {* even and odd for nats *} |
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|
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lemma pos_int_even_equiv_nat_even: "0 \<le> x ==> even x = even (nat x)" |
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by (simp add: even_nat_def) |
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|
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lemma even_nat_product: "even((x::nat) * y) = (even x | even y)" |
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by (simp add: even_nat_def int_mult) |
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138 |
|
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lemma even_nat_sum: "even ((x::nat) + y) = |
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diff
changeset
|
140 |
((even x & even y) | (odd x & odd y))" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
141 |
by (unfold even_nat_def, simp) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
142 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
143 |
lemma even_nat_difference: |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
144 |
"even ((x::nat) - y) = (x < y | (even x & even y) | (odd x & odd y))" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
145 |
apply (auto simp add: even_nat_def zdiff_int [THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
146 |
apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
147 |
apply (case_tac "x < y", auto simp add: zdiff_int [THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
148 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
149 |
|
14436 | 150 |
lemma even_nat_Suc: "even (Suc x) = odd x" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
151 |
by (simp add: even_nat_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
152 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
153 |
lemma even_nat_power: "even ((x::nat)^y) = (even x & 0 < y)" |
16413 | 154 |
by (simp add: even_nat_def int_power) |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
155 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
156 |
lemma even_nat_zero: "even (0::nat)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
157 |
by (simp add: even_nat_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
158 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
159 |
lemmas even_odd_nat_simps [simp] = even_nat_def[of "number_of v",standard] |
14436 | 160 |
even_nat_zero even_nat_Suc even_nat_product even_nat_sum even_nat_power |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
161 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
162 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
163 |
subsection {* Equivalent definitions *} |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
164 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
165 |
lemma nat_lt_two_imp_zero_or_one: "(x::nat) < Suc (Suc 0) ==> |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
166 |
x = 0 | x = Suc 0" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
167 |
by auto |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
168 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
169 |
lemma even_nat_mod_two_eq_zero: "even (x::nat) ==> x mod (Suc (Suc 0)) = 0" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
170 |
apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
171 |
apply (drule subst, assumption) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
172 |
apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
173 |
apply force |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
174 |
apply (subgoal_tac "0 < Suc (Suc 0)") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
175 |
apply (frule mod_less_divisor [of "Suc (Suc 0)" x]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
176 |
apply (erule nat_lt_two_imp_zero_or_one, auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
177 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
178 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
179 |
lemma odd_nat_mod_two_eq_one: "odd (x::nat) ==> x mod (Suc (Suc 0)) = Suc 0" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
180 |
apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
181 |
apply (drule subst, assumption) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
182 |
apply (subgoal_tac "x mod Suc (Suc 0) = 0 | x mod Suc (Suc 0) = Suc 0") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
183 |
apply force |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
184 |
apply (subgoal_tac "0 < Suc (Suc 0)") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
185 |
apply (frule mod_less_divisor [of "Suc (Suc 0)" x]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
186 |
apply (erule nat_lt_two_imp_zero_or_one, auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
187 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
188 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
189 |
lemma even_nat_equiv_def: "even (x::nat) = (x mod Suc (Suc 0) = 0)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
190 |
apply (rule iffI) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
191 |
apply (erule even_nat_mod_two_eq_zero) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
192 |
apply (insert odd_nat_mod_two_eq_one [of x], auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
193 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
194 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
195 |
lemma odd_nat_equiv_def: "odd (x::nat) = (x mod Suc (Suc 0) = Suc 0)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
196 |
apply (auto simp add: even_nat_equiv_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
197 |
apply (subgoal_tac "x mod (Suc (Suc 0)) < Suc (Suc 0)") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
198 |
apply (frule nat_lt_two_imp_zero_or_one, auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
199 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
200 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
201 |
lemma even_nat_div_two_times_two: "even (x::nat) ==> |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
202 |
Suc (Suc 0) * (x div Suc (Suc 0)) = x" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
203 |
apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
204 |
apply (drule even_nat_mod_two_eq_zero, simp) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
205 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
206 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
207 |
lemma odd_nat_div_two_times_two_plus_one: "odd (x::nat) ==> |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
208 |
Suc( Suc (Suc 0) * (x div Suc (Suc 0))) = x" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
209 |
apply (insert mod_div_equality [of x "Suc (Suc 0)", THEN sym]) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
210 |
apply (drule odd_nat_mod_two_eq_one, simp) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
211 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
212 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
213 |
lemma even_nat_equiv_def2: "even (x::nat) = (EX y. x = Suc (Suc 0) * y)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
214 |
apply (rule iffI, rule exI) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
215 |
apply (erule even_nat_div_two_times_two [THEN sym], auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
216 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
217 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
218 |
lemma odd_nat_equiv_def2: "odd (x::nat) = (EX y. x = Suc(Suc (Suc 0) * y))" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
219 |
apply (rule iffI, rule exI) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
220 |
apply (erule odd_nat_div_two_times_two_plus_one [THEN sym], auto) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
221 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
222 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
223 |
subsection {* Parity and powers *} |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
224 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
225 |
lemma minus_one_even_odd_power: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
226 |
"(even x --> (- 1::'a::{comm_ring_1,recpower})^x = 1) & |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
227 |
(odd x --> (- 1::'a)^x = - 1)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
228 |
apply (induct x) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
229 |
apply (rule conjI) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
230 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
231 |
apply (insert even_nat_zero, blast) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
232 |
apply (simp add: power_Suc) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
233 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
234 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
235 |
lemma minus_one_even_power [simp]: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
236 |
"even x ==> (- 1::'a::{comm_ring_1,recpower})^x = 1" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
237 |
by (rule minus_one_even_odd_power [THEN conjunct1, THEN mp], assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
238 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
239 |
lemma minus_one_odd_power [simp]: |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
240 |
"odd x ==> (- 1::'a::{comm_ring_1,recpower})^x = - 1" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
241 |
by (rule minus_one_even_odd_power [THEN conjunct2, THEN mp], assumption) |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
242 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
243 |
lemma neg_one_even_odd_power: |
15003 | 244 |
"(even x --> (-1::'a::{number_ring,recpower})^x = 1) & |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
245 |
(odd x --> (-1::'a)^x = -1)" |
15251 | 246 |
apply (induct x) |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
247 |
apply (simp, simp add: power_Suc) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
248 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
249 |
|
14436 | 250 |
lemma neg_one_even_power [simp]: |
15003 | 251 |
"even x ==> (-1::'a::{number_ring,recpower})^x = 1" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
252 |
by (rule neg_one_even_odd_power [THEN conjunct1, THEN mp], assumption) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
253 |
|
14436 | 254 |
lemma neg_one_odd_power [simp]: |
15003 | 255 |
"odd x ==> (-1::'a::{number_ring,recpower})^x = -1" |
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
256 |
by (rule neg_one_even_odd_power [THEN conjunct2, THEN mp], assumption) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
257 |
|
14443
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14436
diff
changeset
|
258 |
lemma neg_power_if: |
15003 | 259 |
"(-x::'a::{comm_ring_1,recpower}) ^ n = |
14443
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14436
diff
changeset
|
260 |
(if even n then (x ^ n) else -(x ^ n))" |
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14436
diff
changeset
|
261 |
by (induct n, simp_all split: split_if_asm add: power_Suc) |
75910c7557c5
generic theorems about exponentials; general tidying up
paulson
parents:
14436
diff
changeset
|
262 |
|
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
263 |
lemma zero_le_even_power: "even n ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
264 |
0 <= (x::'a::{recpower,ordered_ring_strict}) ^ n" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
265 |
apply (simp add: even_nat_equiv_def2) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
266 |
apply (erule exE) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
267 |
apply (erule ssubst) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
268 |
apply (subst power_add) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
269 |
apply (rule zero_le_square) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
270 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
271 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
272 |
lemma zero_le_odd_power: "odd n ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
273 |
(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = (0 <= x)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
274 |
apply (simp add: odd_nat_equiv_def2) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
275 |
apply (erule exE) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
276 |
apply (erule ssubst) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
277 |
apply (subst power_Suc) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
278 |
apply (subst power_add) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
279 |
apply (subst zero_le_mult_iff) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
280 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
281 |
apply (subgoal_tac "x = 0 & 0 < y") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
282 |
apply (erule conjE, assumption) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
283 |
apply (subst power_eq_0_iff [THEN sym]) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
284 |
apply (subgoal_tac "0 <= x^y * x^y") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
285 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
286 |
apply (rule zero_le_square)+ |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
287 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
288 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
289 |
lemma zero_le_power_eq: "(0 <= (x::'a::{recpower,ordered_idom}) ^ n) = |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
290 |
(even n | (odd n & 0 <= x))" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
291 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
292 |
apply (subst zero_le_odd_power [THEN sym]) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
293 |
apply assumption+ |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
294 |
apply (erule zero_le_even_power) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
295 |
apply (subst zero_le_odd_power) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
296 |
apply assumption+ |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
297 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
298 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
299 |
lemma zero_less_power_eq: "(0 < (x::'a::{recpower,ordered_idom}) ^ n) = |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
300 |
(n = 0 | (even n & x ~= 0) | (odd n & 0 < x))" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
301 |
apply (rule iffI) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
302 |
apply clarsimp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
303 |
apply (rule conjI) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
304 |
apply clarsimp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
305 |
apply (rule ccontr) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
306 |
apply (subgoal_tac "~ (0 <= x^n)") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
307 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
308 |
apply (subst zero_le_odd_power) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
309 |
apply assumption |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
310 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
311 |
apply (rule notI) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
312 |
apply (simp add: power_0_left) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
313 |
apply (rule notI) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
314 |
apply (simp add: power_0_left) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
315 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
316 |
apply (subgoal_tac "0 <= x^n") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
317 |
apply (frule order_le_imp_less_or_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
318 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
319 |
apply (erule zero_le_even_power) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
320 |
apply (subgoal_tac "0 <= x^n") |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
321 |
apply (frule order_le_imp_less_or_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
322 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
323 |
apply (subst zero_le_odd_power) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
324 |
apply assumption |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
325 |
apply (erule order_less_imp_le) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
326 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
327 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
328 |
lemma power_less_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n < 0) = |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
329 |
(odd n & x < 0)" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
330 |
apply (subst linorder_not_le [THEN sym])+ |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
331 |
apply (subst zero_le_power_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
332 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
333 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
334 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
335 |
lemma power_le_zero_eq: "((x::'a::{recpower,ordered_idom}) ^ n <= 0) = |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
336 |
(n ~= 0 & ((odd n & x <= 0) | (even n & x = 0)))" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
337 |
apply (subst linorder_not_less [THEN sym])+ |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
338 |
apply (subst zero_less_power_eq) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
339 |
apply auto |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
340 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
341 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
342 |
lemma power_even_abs: "even n ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
343 |
(abs (x::'a::{recpower,ordered_idom}))^n = x^n" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
344 |
apply (subst power_abs [THEN sym]) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
345 |
apply (simp add: zero_le_even_power) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
346 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
347 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
348 |
lemma zero_less_power_nat_eq: "(0 < (x::nat) ^ n) = (n = 0 | 0 < x)" |
18648 | 349 |
by (induct n, auto) |
16775
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
350 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
351 |
lemma power_minus_even [simp]: "even n ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
352 |
(- x)^n = (x^n::'a::{recpower,comm_ring_1})" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
353 |
apply (subst power_minus) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
354 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
355 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
356 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
357 |
lemma power_minus_odd [simp]: "odd n ==> |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
358 |
(- x)^n = - (x^n::'a::{recpower,comm_ring_1})" |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
359 |
apply (subst power_minus) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
360 |
apply simp |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
361 |
done |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
362 |
|
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
363 |
(* Simplify, when the exponent is a numeral *) |
c1b87ef4a1c3
added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents:
16413
diff
changeset
|
364 |
|
17085 | 365 |
lemmas power_0_left_number_of = power_0_left [of "number_of w", standard] |
366 |
declare power_0_left_number_of [simp] |
|
367 |
||
368 |
lemmas zero_le_power_eq_number_of = |
|
369 |
zero_le_power_eq [of _ "number_of w", standard] |
|
370 |
declare zero_le_power_eq_number_of [simp] |
|
371 |
||
372 |
lemmas zero_less_power_eq_number_of = |
|
373 |
zero_less_power_eq [of _ "number_of w", standard] |
|
374 |
declare zero_less_power_eq_number_of [simp] |
|
375 |
||
376 |
lemmas power_le_zero_eq_number_of = |
|
377 |
power_le_zero_eq [of _ "number_of w", standard] |
|
378 |
declare power_le_zero_eq_number_of [simp] |
|
379 |
||
380 |
lemmas power_less_zero_eq_number_of = |
|
381 |
power_less_zero_eq [of _ "number_of w", standard] |
|
382 |
declare power_less_zero_eq_number_of [simp] |
|
383 |
||
384 |
lemmas zero_less_power_nat_eq_number_of = |
|
385 |
zero_less_power_nat_eq [of _ "number_of w", standard] |
|
386 |
declare zero_less_power_nat_eq_number_of [simp] |
|
387 |
||
388 |
lemmas power_eq_0_iff_number_of = power_eq_0_iff [of _ "number_of w", standard] |
|
389 |
declare power_eq_0_iff_number_of [simp] |
|
390 |
||
391 |
lemmas power_even_abs_number_of = power_even_abs [of "number_of w" _, standard] |
|
392 |
declare power_even_abs_number_of [simp] |
|
393 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
394 |
|
17472 | 395 |
subsection {* An Equivalence for @{term [source] "0 \<le> a^n"} *} |
14450 | 396 |
|
397 |
lemma even_power_le_0_imp_0: |
|
15003 | 398 |
"a ^ (2*k) \<le> (0::'a::{ordered_idom,recpower}) ==> a=0" |
14450 | 399 |
apply (induct k) |
400 |
apply (auto simp add: zero_le_mult_iff mult_le_0_iff power_Suc) |
|
401 |
done |
|
402 |
||
403 |
lemma zero_le_power_iff: |
|
15003 | 404 |
"(0 \<le> a^n) = (0 \<le> (a::'a::{ordered_idom,recpower}) | even n)" |
14450 | 405 |
(is "?P n") |
406 |
proof cases |
|
407 |
assume even: "even n" |
|
14473 | 408 |
then obtain k where "n = 2*k" |
14450 | 409 |
by (auto simp add: even_nat_equiv_def2 numeral_2_eq_2) |
410 |
thus ?thesis by (simp add: zero_le_even_power even) |
|
411 |
next |
|
412 |
assume odd: "odd n" |
|
14473 | 413 |
then obtain k where "n = Suc(2*k)" |
14450 | 414 |
by (auto simp add: odd_nat_equiv_def2 numeral_2_eq_2) |
415 |
thus ?thesis |
|
416 |
by (auto simp add: power_Suc zero_le_mult_iff zero_le_even_power |
|
417 |
dest!: even_power_le_0_imp_0) |
|
418 |
qed |
|
419 |
||
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
420 |
subsection {* Miscellaneous *} |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
421 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
422 |
lemma even_plus_one_div_two: "even (x::int) ==> (x + 1) div 2 = x div 2" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
423 |
apply (subst zdiv_zadd1_eq) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
424 |
apply (simp add: even_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
425 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
426 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
427 |
lemma odd_plus_one_div_two: "odd (x::int) ==> (x + 1) div 2 = x div 2 + 1" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
428 |
apply (subst zdiv_zadd1_eq) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
429 |
apply (simp add: even_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
430 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
431 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
432 |
lemma div_Suc: "Suc a div c = a div c + Suc 0 div c + |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
433 |
(a mod c + Suc 0 mod c) div c" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
434 |
apply (subgoal_tac "Suc a = a + Suc 0") |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
435 |
apply (erule ssubst) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
436 |
apply (rule div_add1_eq, simp) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
437 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
438 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
439 |
lemma even_nat_plus_one_div_two: "even (x::nat) ==> |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
440 |
(Suc x) div Suc (Suc 0) = x div Suc (Suc 0)" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
441 |
apply (subst div_Suc) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
442 |
apply (simp add: even_nat_equiv_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
443 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
444 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
445 |
lemma odd_nat_plus_one_div_two: "odd (x::nat) ==> |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
446 |
(Suc x) div Suc (Suc 0) = Suc (x div Suc (Suc 0))" |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
447 |
apply (subst div_Suc) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
448 |
apply (simp add: odd_nat_equiv_def) |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
449 |
done |
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
450 |
|
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
diff
changeset
|
451 |
end |